Jose Ramon Real - cp's OEIS frontend

User: Jose Ramon Real

Jose Ramon Real has authored 4 sequences.

A145324 Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....

1, 1, 2, 1, 5, 6, 1, 9, 26, 24, 1, 14, 71, 154, 120, 1, 20, 155, 580, 1044, 720, 1, 27, 295, 1665, 5104, 8028, 5040, 1, 35, 511, 4025, 18424, 48860, 69264, 40320, 1, 44, 826, 8624, 54649, 214676, 509004, 663696, 362880, 1, 54, 1266, 16884, 140889, 761166

Offset: 1

Jose Ramon Real, Oct 07 2008

The last number of row n is n!.
Essentially the triangle given by [1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [2, 1, 3, 2, 4, 3, 5, 4, 6, 5, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 09 2008
T(n+1,k+1) = a_k(2,3,...,n+1), n >= 0, k = 0..n, with the elementary symmetric function a_k(x[1],x[2],...,x[n]), with a_0(0):=1. E.g., a_2(2,3,4) = 2*3 + 2*4 + 3*4 = 26 = T(4,3). - Wolfdieter Lang, Oct 24 2011

			From _Wolfdieter Lang_, Oct 24 2011: (Start)
n\k 1  2   3    4     5    6     7 ...
1:  1
2:  1  2
3:  1  5   6
4:  1  9  26   24
5:  1 14  71  154   120
6:  1 20 155  580  1044  720
7:  1 27 295 1665  5104 8028  5040
...
T(4,3)= 26 = |s(5,3)| - |s(5,4)| + |s(5,5)| = 35 - 10 + 1.
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened)
Olivier Bodini, Antoine Genitrini, Mehdi Naima, Ranked Schröder Trees, arXiv:1808.08376 [cs.DS], 2018.
Olivier Bodini, Antoine Genitrini, Cécile Mailler, Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]

A145324 := proc(n,k) coeftayl( 1*mul(x+i,i=2..n),x=0,n-k) ; end: for n from 1 to 11 do for k from 1 to n do printf("%d,",A145324(n,k)) ; od: od: # R. J. Mathar, Oct 10 2008

Table[Reverse[CoefficientList[Product[x+j, {j, 2, k}], x]], {k, 1, 15}] // Flatten (* Robert A. Russell, Sep 29 2018 *)

T(n,k) = A143491(n+1,n+2-k). - R. J. Mathar, Oct 10 2008
T(n,k) = Sum_{m=0..k-1} (-1)^m*|s(n+1, n+2-k+m)|, n >= 1, k = 1..n, with the Stirling numbers of the first kind s(n,k) = A048994(n,k). - Wolfdieter Lang, Oct 24 2011
T(n,k) = T(n-1,k)+n*T(n-1,k-1). - Mikhail Kurkov, Jun 26 2018

More terms from R. J. Mathar, Oct 10 2008

A135299 Pascal's triangle, but the last element of the row is the sum of all the previous terms.

Original entry on oeis.org

1, 1, 2, 1, 3, 8, 1, 4, 11, 32, 1, 5, 15, 43, 128, 1, 6, 20, 58, 171, 512, 1, 7, 26, 78, 229, 683, 2048, 1, 8, 33, 104, 307, 912, 2731, 8192, 1, 9, 41, 137, 411, 1219, 3643, 10923, 32768, 1, 10, 50, 178, 548, 1630, 4862, 14566, 43691, 131072

Offset: 0

Jose Ramon Real, Dec 04 2007

			T(2,1) = T(1,0) + T(1,1) = 1 + 2 = 3;
T(2,2) = T(0,0) + T(1,0) + T(1,1) + T(2,0) + T(2,1) = 1 + 1 + 2 + 1 + 3 = 8.
From _G. C. Greubel_, Oct 09 2016: (Start)
The triangle is:
  1;
  1, 2;
  1, 3,  8;
  1, 4, 11, 32;
  1, 5, 15, 43, 128;
  1, 6, 20, 58, 171, 512;
  ... (End)

G. C. Greubel, Table of n, a(n) for the first 25 rows

Cf. A007318, A067337.

T[0, 0] := 1; T[n_, 0] := 1; T[n_, k_] := T[n - 1, k] + T[n - 1, k - 1]; T[n_, n_] := 2^(2*n - 1); Table[T[n, k], {n, 0, 5}, {k, 0, n}] (* G. C. Greubel, Oct 09 2016 *)

T(0,0) = 1;
T(n,0) = 1;
T(n,k) = T(n-1, k-1) + T(n-1, k) if k < n;
T(n,n) = (Sum_{j=0..n-1} Sum_{i=0..j} T(j,i)) + Sum_{i=0..n-1} T(n,i) [i.e., sum of all earlier terms of the triangle].
T(n,n) = (4^n)/2 for n > 0;
T(n,n) = 2*Sum_{i=0..n-1} T(n,i).

A131183 a(n) = a(n-1) + a(n-2) if n == 3 mod 4; a(n) = a(n-1) - a(n-2) if n == 0 mod 4; a(n) = a(n-1)*a(n-2) if n == 1 mod 4; and a(n) = a(n-1)/a(n-2) if n == 2 mod 4; with a(1)=a(2)=1.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 4, 2, 8, 4, 12, 8, 96, 12, 108, 96, 10368, 108, 10476, 10368, 108615168, 10476, 108625644, 108615168, 11798392572168192, 108625644, 11798392680793836, 11798392572168192, 139202068568601556987554268864512

Offset: 1

Jose Ramon Real, Oct 22 2007

If S(n)=a(4n-1) (i.e., term "+"), R(n)=a(4n) (i.e., "-"), P(n)=a(4n+1), D(n)=a(4n+2) then D(n)=S(n), P(n)=S(n+1)-S(n), R(n+1)=P(n)=S(n+1)-S(n). - Jose Ramon Real, Nov 10 2007

			a(3) = a(2) + a(1) = 1 + 1 = 2;
a(4) = a(3) - a(2) = 2 - 1 = 1;
a(5) = a(4) * a(3) = 1 * 2 = 2;
a(6) = a(5) / a(4) = 2 / 1 = 2.

A131183 := proc(n) option remember ; if n <= 2 then 1 ; elif n mod 4 = 3 then A131183(n-1)+A131183(n-2) ; elif n mod 4 = 0 then A131183(n-1)-A131183(n-2) ; elif n mod 4 = 1 then A131183(n-1)*A131183(n-2) ; else A131183(n-1)/A131183(n-2) ; fi ; end: seq(A131183(n),n=1..35) ; # R. J. Mathar, Oct 28 2007

a[1]=a[2]=1; a[n_] := a[n] = Switch[Mod[n, 4], 3, a[n-1]+a[n-2], 0, a[n-1]-a[n-2], 1, a[n-1]*a[n-2], 2, a[n-3]]; Array[a, 30] (* Jean-François Alcover, Dec 28 2015 *)
nxt[{n_,a_,b_}]:=Module[{m=Mod[n+1,4]},{n+1,b,Which[m==3,a+b,m==0,b-a, m==1,a*b,m==2,b/a]}]; Join[{1,1,2},NestList[nxt,{1,1,2},30][[All,2]]] (* Harvey P. Dale, Sep 04 2017 *)

More terms from R. J. Mathar, Oct 28 2007

A128920 Sum of all the factors in all the ways to write n as n = xyz with 1 <= x <= y <= z <= n.

Original entry on oeis.org

3, 4, 5, 11, 7, 14, 9, 23, 18, 20, 13, 38, 15, 26, 26, 46, 19, 50, 21, 54, 34, 38, 25, 83, 38, 44, 51, 70, 31, 86, 33, 88, 50, 56, 50, 136, 39, 62, 58, 119, 43, 112, 45, 102, 92, 74, 49, 181, 66, 108, 74, 118, 55, 150, 74, 155, 82, 92, 61, 233, 63, 98, 120, 185, 86, 164, 69

Offset: 1

Jose Ramon Real, Oct 23 2007

nonn

			a(6)=14 because 6 = 6*1*1 = 3*2*1 and 6+1+1+3+2+1 = 14.

Cf. A034836.

Corrected by Charles R Greathouse IV, Sep 02 2009

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User: Jose Ramon Real

A145324 Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....

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A135299 Pascal's triangle, but the last element of the row is the sum of all the previous terms.

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A131183 a(n) = a(n-1) + a(n-2) if n == 3 mod 4; a(n) = a(n-1) - a(n-2) if n == 0 mod 4; a(n) = a(n-1)*a(n-2) if n == 1 mod 4; and a(n) = a(n-1)/a(n-2) if n == 2 mod 4; with a(1)=a(2)=1.

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A128920 Sum of all the factors in all the ways to write n as n = x*y*z with 1 <= x <= y <= z <= n.

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A128920 Sum of all the factors in all the ways to write n as n = xyz with 1 <= x <= y <= z <= n.